K.G. Kamalutdinov. Self-intersections in parametrized self-similar sets under translations and extensions of copies ... P. 116-124

We study the problem of pairwise intersections $F_i(K_t)\cap F_j^t (K_t)$ of different copies of a self-similar set $K_t$ generated by a system $\mathcal F_t=\{F_1,\dots,F_m\}$ of contracting similarities in $\mathbb R^n$, where one mapping $F_j^t$ depends on a real or vector parameter $t$. Two cases are considered: the parameter $t\in \mathbb R^n$ specifies a translation of a mapping $F_j^t(x) = G(x)+t$, and the parameter $t\in (a,b)$ is the similarity coefficient of a mapping $F_j^t(x)=tG(x)+h$, where $0<a<b<1$ and $G$ is an isometry of $\mathbb R^n$. We impose some constraints on the similarity coefficients of mappings of the system $\mathcal F_t$ and require that the similarity dimension of the system does not exceed some number~$s$. For such systems it is proved that the Hausdorff dimension of the set of parameters $t$ for which the intersection $F_i(K_t)\cap F_j^t(K_t)$ is nonempty does not exceed $2s$. The obtained results are applied to the problem of checking the strong separation condition for a system $\mathcal F_\tau=\{F_1^\tau,\dots, F_m^\tau\}$ of contraction similarities depending on a parameter vector $\tau=(t_1,\dots,t_m)$. Two cases are considered: $\tau$ is a vector of translations of mappings $F_i^\tau(x)=G_i(x)+t_i$, $t_i\in \mathbb R^n$, and $\tau$ is a vector of similarity coefficients of mappings $F_i^\tau(x)=t_i G_i(x)+h_i$, $t_i\in(a,b)$, where $0<a<b<1$ and all $G_i$ are isometries in $\mathbb R^n$. In both cases we find sufficient conditions for the system $\mathcal F_\tau$ to satisfy the strong separation condition for almost all values of $\tau$. We also consider the easier problem of the intersection $A\cap f_t(B)$ for a pair of compact sets $A$ and $B$ in the space $\mathbb R^n$. Two cases are considered: $f_t(B)=B+t$ for $t\in\mathbb R^n$, and $f_t(B)=tB$ for $t\in\mathbb R$, where the closure of $B$ does not contain the origin. In both cases it is proved that the Hausdorff dimension of the set of parameters $t$ for which the intersection $A\cap f_t(B)$ is nonempty does not exceed $\dim_H (A\times B)$. Consequently, when the dimension of the product $A\times B$ is small enough, the empty intersection $A\cap f_t(B)$ is guaranteed for almost all values of $t$ in both cases.

Keywords: self-similar fractal, general position, strong separation condition, Hausdorff dimension

Received March 22, 2019

Revised May 6, 2019

Accepted May 13, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research (projects no. 19-01-00569, 18-501-51021).

Kirill Glebovich Kamalutdinov, Novosibirsk State University, Novosibirsk, 630090 Russia,
e-mail: kirdan15@mail.ru

REFERENCES

1.   Hutchinson J. Fractals and self-similarity. Indiana Univ. Math. J., 1981, vol. 30, no. 5, pp. 713–747. doi: 10.1512/iumj.1981.30.30055 

2.   Marstrand J.M. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc., 1954, vol. s3–4, no. 1, pp. 257–302. doi: 10.1112/plms/s3-4.1.257 

3.   Mattila P. Hausdorff dimension and capacities of intersections of sets in $n$-space. Acta Math., 1984, vol. 152, pp. 77–105. doi: 10.1007/BF02392192 

4.   Falconer K.J. Dimensions of intersections and distance sets for polyhedral norms. Real Anal. Exchange, 2004, vol. 30, no. 2, pp. 719–726. doi: 10.14321/realanalexch.30.2.0719 

5.   Pollicott M., Simon K. The Hausdorff dimension of $\lambda$-expansions with deleted digits. Trans. Am. Math. Soc., 1995, vol. 347, no. 3, pp. 967–983. doi: 10.2307/2154881 

6.   Kamalutdinov K.G., Tetenov A.V. Twofold Cantor sets in $\mathbb R$. Sib. Elektron. Mat. Izv., 2018, vol. 15, pp. 801–814. doi: 10.17377/semi.2018.15.066 

7.   Falconer K.J. Fractal geometry: mathematical foundations and applications. 3rd ed. New York: J. Wiley and Sons, 2014, 398 p. ISBN: 9781118762851 

8.   Edgar G. Measure, Topology, and Fractal Geometry. 2nd ed. N Y: Springer-Verlag, 2008, 272 p. doi: 10.1007/978-0-387-74749-1 

9.   Kuratowski K. Topology. Vol. II. N Y; London: Acad. Press, 1968, 608 p. ISBN: 978-0-12-429202-4 . Translated to Russian under the title Topologiya. T. 2. Moscow, Mir Publ., 1969, 624 p.

Cite this article as: K.G.Kamalutdinov. Self-intersections in parametrized self-similar sets under translations and extensions of copies, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 116–124.